Parameter estimation of discrete-time sinusoidal signals: A nonlinear control approach

Abstract

This paper considers a parameter estimation problem for discrete-time sinusoidal signals in the presence of measurement noise. Different from most of existing methods, we shall solve the problem based on nonlinear control theory. Specifically, we introduce a constructive nonlinear estimator that enables us to prove an interesting input/output stability property from measurement noise to estimation error. To show efficacy of the proposed method, we also present several numerical examples using different types of signals and noises.

Introduction

Parameter estimation of sinusoidal signals is fundamental in engineering with broad applications in the fields such as communication, image processing, power engineering, and automatic control, see, e.g., Kay (1988), Kay and Marple (1981) and Quinn and Hannan (2001) for overview. In literature, the problem has been studied for many years and typically handled by statistical methods, see, e.g., Bittanti and Savaresi (2000), Kay (1989), Quinn (1994), Rife and Boorstyn (1974), So and Chan (2006), Stoica, Li, and Li (2000), Stoica, Moses, Friedlander, and Söderström (1989) and Tretter (1985). Some other methods include, see, e.g., subspace methods (Paulraj et al., 1986, Schmidt, 1986) and compressed sensing methods (Candès and Fernandez-Granda, 2014, Tang et al., 2013, Yang et al., 2018).

Different from the aforementioned methods, a recent trend is to tackle the problem based on nonlinear control theory, see, e.g., Carnevale and Astolfi (2012), Chen, Pin, Ng, Hui, and Parisini (2018), Hou (2012), Na, Yang, Wu, and Guo (2015) and Tedesco, Casavola, and Fedele (2017). The works of Carnevale and Astolfi (2012), Chen et al. (2018), Hou (2012) and Na et al. (2015) consider the parameter estimation problem of continuous-time sinusoidal signals. In particular, Hou (2012) proposed, based on the result in Xia (2002), a first solution to estimate parameters (offset, frequencies, amplitudes, and phases) of a biased multi-sinusoidal signal and showed that the estimation error has the global asymptotic convergence property. Tedesco et al. (2017) proposed an interesting method for parameter estimation of a discrete-time single-sinusoidal signal without measurement noise, namely discrete-time frequency-locked-loop (in short, FLL) nonlinear filters. It can be viewed as a distinguished discrete-time investigation of the continuous-time FLL filter addressed in Fedele, Ferrise, and Frascino (2009). It is pointed out that in general, parameter estimation of discrete-time sinusoidal signals would not be a straight extension of their continuous-time counterparts, because of, for example, the digitalization feature and challenges in stability analysis of discrete-time systems.

The present study is to investigate a general parameter estimation problem of discrete-time multi-sinusoidal signals in the presence of measurement noise. Particularly, we explore a robustness analysis of the estimation system. Toward this end, we first convert the problem to a state/parameter estimation problem of an unknown linear time-invariant system. Specifically, we start with the special noise-free case by means of constructing an estimator when both the state and system parameter are available and constructing a filter to estimate the state/parameter of the modeled system in question. It leads to a nonlinear dynamical estimator (in short, NDE) of state-space-model-based, consisting of the constructed estimator and dynamical filter. It can achieve parameter estimation of multi-sinusoidal signals. Particularly, it enables us to carry out a careful robustness analysis of the proposed NDE method based on an input/output stability perspective with a concern of the affects of the measurement noise. Hence, in comparison with Tedesco et al. (2017), the present study is not only a strictly extended study for general biased multiple sinusoids, but also to provide an interesting robustness analysis.

The rest of this paper is organized as follows. Section 2 formulates the parameter estimation problems. Section 3 focuses on the basic noise-free case, and Section 4 considers the noisy case and the relevant robustness analysis. Section 5 gives numerical simulations. Finally, Section 6 closes the paper.

Notation: $\mathbb{N}$ (or ${\mathbb{R}}_{\ge 0}$) is the set of nonnegative integers (or real numbers), and ${\mathbb{N}}_{\ge i}$ is the set of nonnegative integers that greater than or equal to $i$. $\left|\cdot \right|$ is the absolute value of a real number. $\Vert \cdot \Vert$ is the Euclidean norm or the induced matrix norm. ${\Vert \cdot \Vert }_{\infty }$ is the essential norm, i.e., for a function $u:\mathbb{N}\to {\mathbb{R}}^n$, ${\Vert u\Vert }_{\infty }=sup\{\Vert u\left(k\right)\Vert :k\in \mathbb{N}\}$. For vector $x\in {\mathbb{R}}^n$ and matrix $P\in {\mathbb{R}}^{n\times n}$, $x_i$ denotes the $i$th entry of $x$ and ${\Vert x\Vert }_P≔x^{\mathsf{T}}Px$. $I_n$ denotes the $n$ dimensional identity matrix. $L_{\infty }$ denotes the set of time-varying functions $u\left(k\right)$ satisfying ${\Vert u\Vert }_{\infty }<\infty$. ${\left\{x_i\right\}}_{i=1}^n$ is the set of vectors $x_1,\dots ,x_n$. $\mathcal{P}$ is the set of positive definite functions. $\mathcal{K}$ is the set of continuous, positive definite, and strictly increasing functions $\alpha :{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$. ${\mathcal{K}}_{\infty }$ is the set of unbounded functions in $\mathcal{K}$. $\mathcal{K}\mathcal{L}$ is the set of continuous functions $\beta :{\mathbb{R}}_{\ge 0}\times {\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ satisfying: for each fixed $t\ge 0$, $\beta (\cdot ,t)\in \mathcal{K}$; and for each fixed $s$, $\beta (s,\cdot )$ is decreasing and ${lim}_{k\to \infty }\beta (s,k)=0$. For a sequence ${\left\{p_k\right\}}_{k=0}^{\infty }$, ${lim\; sup}_{k\to \infty }{p}_k≔{lim}_{k\to \infty }\left({sup}_{i\ge k}{p}_i\right)$. For a real square matrix $A$, $\lambda \left(A\right)$ collects all the eigenvalues of $A$, and $det\left(A\right)$ is its determinant. We sometimes use the shorthand notation

$$\mathbf{sc}\left(s\right)≔{(sin\left(s\right),cos\left(s\right))}^{\mathsf{T}},\mathbf{cs}\left(s\right)≔{(cos\left(s\right),sin\left(s\right))}^{\mathsf{T}},s\in \mathbb{R}.$$